-csc(x)=1

{ "query": { "display": "$$-\\csc\\left(x\\right)=1$$", "symbolab_question": "EQUATION#-\\csc(x)=1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{3π}{2}+2πn", "degrees": "x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$-\\csc\\left(x\\right)=1{\\quad:\\quad}x=\\frac{3π}{2}+2πn$$", "input": "-\\csc\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-\\csc\\left(x\\right)=1", "result": "\\csc\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-\\csc\\left(x\\right)}{-1}=\\frac{1}{-1}" }, { "type": "step", "primary": "Simplify", "result": "\\csc\\left(x\\right)=-1" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TIbpFXISs0C1OtEg88W6Vc+Pqip+BMTSVx/OhjiEJApPnie/F7TDTqTxkucfXTe/aPjNWQqta/TPfIU5KdkDBWZK5M8/+KgAvP2E+DfEo/tlR2ULm1rtqmM+yHvMQ9ujT6E+Z+lYFfRqAnQ26sDvnCwtYv+gYP75aLTZcT7K4J2HjCJOLglkInccUWJVHAEMOkr2eGdxDqvMbGz01KDfKEZVwY9fmtqcF1NEPhJii2en6WU7ybUlSwezFhLKvkKNVvuqUoYkKmBK6fSR2cNlgkfqfT3HSxldmNt6rUcfvySCyrAO9zMQDZyjEiWM7qI5HNLamESyiXkEVNU4Be3yXlX3rP7J381OCLdf7C7WWRDvbBmbuQNTF0TphKZ8RuvaTbBiSexc/wbzzjAmcOEO4P2EarIp30Zm4kqsvvvDubU=" } }, { "type": "interim", "title": "General solutions for $$\\csc\\left(x\\right)=-1$$", "result": "x=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\csc\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\csc(x)&x&\\csc(x)\\\\\\hline 0&\\mathrm{Undefiend}&π&\\mathrm{Undefiend}\\\\\\hline \\frac{π}{6}&2&\\frac{7π}{6}&-2\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&\\frac{2\\sqrt{3}}{3}&\\frac{4π}{3}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{2\\sqrt{3}}{3}&\\frac{5π}{3}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{3π}{4}&\\sqrt{2}&\\frac{7π}{4}&-\\sqrt{2}\\\\\\hline \\frac{5π}{6}&2&\\frac{11π}{6}&-2\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions csc 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-\\csc(x)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }